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一阶常系数齐次线性微分方程

$$y ^{\prime} + a y \ {=} \ 0,\ a ^{\prime} \ {=} \ 0$$

$\Gamma: \left\langle y,\ a \right\rangle$

$\mathbf{Solution}\quad \alpha_0: x,\ x \Gamma y \quad \square \to \Gamma_0 \to \Gamma_1$

$\Gamma \to \Gamma_0 \ {=} \ \Gamma \mid \alpha_0$

改写导数形式

$${\color{Teal} \frac{\text{d} y}{\text{d} x}} + a {\color{Gray} y} \ {=} \ 0$$

$\left[\mathbb R :: \text{div}\right]$

$\alpha_1: \left\langle y \ne 0 \right\rangle \quad \square \to \Gamma_1$

$\_ : \left\langle y \ {=} \ 0 \right\rangle \quad \blacksquare$

$\Gamma \to \Gamma_0 \to \Gamma_1 \ {=} \ \Gamma_0 \mid \alpha_1$

除以 $y$

$${\color{Teal} \frac{1}{y}} \frac{\text{d} y }{\text{d} x} + a \ {=} \ 0$$

对 $x$ 积分

$${\color{Teal} \int {\color{Cyan} \text{d} x} \left\{ {\color{Black} \frac{1}{y}} {\color{Gray} \frac{\text{d} y}{\text{d} x}}\right\}} + a {\color{Teal} \int \text{d} x} \ {=} \ 0$$

$${\color{Grey} \int} {\color{Cyan} \text{d} y} {\color{Gray} \left\{ \frac{1}{y} \right\}} + a {\color{Gray} \int \text{d} x} \ {=} \ 0$$

$\left[\mathbb C :: \text{int}\right]\quad C$

$\mathbf{note}:$ 此处 $\displaystyle \int \frac{\text{d}y}{y}$ 理解为复对数的一个分支，记为 $\ln y$，不同分支相差 $2\pi i$。取指数后多值性消失，故可任选一分支。

$${\color{Cyan} \ln} {\color{Teal} y} + a {\color{Teal} x} \ {=} \ {\color{Teal} C}$$

取指数

$${\color{Gray} y} \cdot {\color{Teal} e} ^{a x} \ {=} \ {\color{Teal} e} ^C$$

$\mathbf{overload}\quad {\color{Peach} C} : C \ne 0$

$$y \ {=} \ {\color{Teal} C} e ^{- a x}$$

$\blacksquare$

$\Gamma_0 \rightleftharpoons \Gamma_1$

$\mathbf{overload}\quad {\color{Peach} C}$

$$y \ {=} \ {\color{Teal} C} e ^{- a x}$$

$\blacksquare$